Clinical protocols may require rescue medication when a patient's condition crosses a prespecified threshold. The outcome under continued non-rescue is then unobserved after first crossing, preventing point identification without extrapolation assumptions. In this paper, we develop sharp partial identification and inference for this hypothetical estimand while retaining the original no-rescue intervention. We show that bounding this mean reduces exactly to restricting conditional mean effects of withholding rescue at first-crossing histories. Building on this representation, restrictions on prespecified crossing-stratum averages yield three results. (i) A finite convex program gives sharp bounds without modeling post-crossing trajectories, attained by full-data laws that preserve the observed distribution. (ii) Confidence intervals incorporate sampling uncertainty and numerical error to cover the entire sharp set, including strata with no observed crossings. (iii) Under local feasibility conditions, sampling and numerical errors control endpoint accuracy even with zero-probability strata, whereas valid coverage requires no such conditions. Our experiments show that the stratum-average restrictions can yield narrower confidence intervals in larger samples, and that accounting for uncertainty in empty strata maintains valid coverage when rescue is rare.
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